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I thought $\log(n)$ was like $100^x = n$ and $\ln(n)$ was $e^x = n$. But when I do $\ln(80)$, it gives me the answer for $\log$. Why is that?

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Because Alpha tries to be compatible with the syntax of most softwares that may use $\log$ as well as $\ln$. If you want the decimal logarithm try $\log10(x)$. – Raymond Manzoni Mar 3 '12 at 15:02
up vote 7 down vote accepted

It's simply a matter of definitions.

In all fields, $\ln$ means the natural log, or log base $e$, so that $\ln n = x$ whenever $e^x = n$. In engineering (and high school), $\log$ usually means the common log, or log base $10$, so that $\log n = x$ whenever $10^x = n$.

However, it happens that in higher mathematics, the common log just isn't very important. So for convenience, mathematicians often use the notation $\log$ to represent the natural log. Wolfram Alpha does things the same way.

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"engineering (and high school)" - lol – Salech Alhasov Mar 3 '12 at 20:46

In theoretical mathematics and in many programming languages "$\log$" usually means natural (base-$e$) logarithm. That's the only logarithm that's important for most theoretical purposes.

Sometimes I've written "log" in an expression on a blackboard and a student has asked "Do you mean logarithm, or natural logarithm?", and I've said "Yes". I dislike "ln" because I suspect it of causing some people to think a natural logarithm is something different from a logarithm, as evidenced by questions like that one.

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Wikipedia note: 'Some mathematicians disapprove of this notation. In his 1985 autobiography, Paul Halmos criticized what he considered the "childish ln notation," which he said no mathematician had ever used. The notation was invented by Irving Stringham, a mathematician.' – endolith May 5 '13 at 20:10
I seem to recall that I'm the author of that Wikipedia passage, or at least the first sentence. The opinion by Halmos is from his autobiography. – Michael Hardy May 5 '13 at 20:42

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